Construction of Iso-contours, Bisectors and Voronoi Diagrams on ...

More documents

Recommendations

Info

IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, VOL. 33, NO. 8, PAGE 1502-1517, AUGUST 2011 8is the distance field value difference between neighborsnb i <strong>and</strong> v. Finally the elements in a reduced vertex listL c are the extracted critical points.5.1.1 Models with Single-Piece <strong>Iso</strong>-ContourIf a triangulated surface M has no critical points withindices 1 or 3, by Property 1, any iso-contour on it is a s-ingle closed curve. To explicitly construct the iso-contourwith any prescribed value on this kind <strong>of</strong> models, wepreprocess the model as follows. Given a source pointp, we compute the geodesic distance field on M <strong>and</strong>find the farthest point q on M. Then we construct ageodesic path from p to q. The path goes through a set<strong>of</strong> triangles: each <strong>of</strong> them covers a distance interval <strong>and</strong>all the intervals in them are continuous in t<strong>and</strong>em. Wesort these triangles into a binary tree indexed by thedistance interval. To construct a particular iso-contour,we search the binary tree <strong>and</strong> find the triangle whosedistance interval contains the iso-contour value.Given a starting triangle, the algorithm runs in amarching process. Without loss <strong>of</strong> generality, we assumethat the iso-contour value is zero. Given the first trianglethrough which the iso-contour passes, we trace the isocontourusing the edge which has the opposite signsat its two vertices. Each such edge guides the isocontourfrom one triangle to another triangle. For eachmarched triangle, the inside iso-contour is determinedby Property 6. If the iso-contour goes through a vertexv <strong>of</strong> M, the 1-ring neighbor triangles <strong>of</strong> v is checkedto find the next triangle to be marched. Our marchingmethod is topology-oriented, since we first check thesign <strong>of</strong> vertices <strong>of</strong> each marched triangle. So our methodis robust to degeneracy <strong>and</strong> is topologically consistent.Property 7: The iso-contour construction algorithmfor preprocessed models with single-piece iso-<strong>contours</strong>takes O(log n ′ + k) time, where k is the number <strong>of</strong>triangles through which the iso-contour passes.To see the above property, note that the geodesic path<strong>and</strong> the iso-contour both have the O(n ′ ) complexity inpreprocessed models. So searching the binary tree takesO(log n ′ ) time to identify the starting triangle <strong>and</strong> themarching process takes O(k) time. Finally the overallcomplexity is O(log n ′ + k).5.1.2 Arbitrary Genus-r (r ≥ 0) ModelsThe algorithm for arbitrary genus-r models is morecomplicated than that for models studied in Section 5.1.1.We preprocess the model as follows. We compute thegeodesic distance field with a prescribed source p on M.Each triangle in M covers a distance interval. We sortall triangles in M into an interval tree [13], [42] indexedby the distance interval <strong>of</strong> each triangle.The genus-r iso-contour algorithm also runs in amarching process. Given a particular iso-contour value,we search the interval tree for stabbing query <strong>and</strong>sort all the triangles, whose distance interval coversthe iso-contour value into a queue Que. The followingalgorithm reports the inquired iso-contour with thenumber <strong>of</strong> closed curves.Algorithm 1: genus_r_iso<strong>contours</strong>(M, c)Input. An iso-contour value c, a preprocessed mesh Mwith constructed distance field <strong>and</strong> interval tree.Output. The requested iso-contour with the number <strong>of</strong>closed curves.1. Stabbing query in the interval tree <strong>and</strong> outputthe result in a queue Que;2. Set curve number cn = 0;3. While (Que is not empty)3.1. cn = cn + 1;3.2. Pop the first element t <strong>of</strong> Que;3.3. Marching triangles with the initial triangle t;3.4. Remove all the marched triangles from Que.Property 8: Algorithm 1 for arbitrary genus-r modelstakes O(log n ′ + k log k) time, where k is the number <strong>of</strong>triangles through which the iso-contour passed.We sketch the pro<strong>of</strong> <strong>of</strong> Property 8 as follows. Aninterval tree for a set <strong>of</strong> n ′ intervals reports all intervalsthat contain a query point in O(log n ′ + k) time, wherek is the number <strong>of</strong> reported intervals [13], [42]. The Quecan be built in O(k) time <strong>and</strong> removing an element witha specified key from Que takes O(log k) time [12]. SoAlgorithm 1 runs in O(log n ′ + k log k) time.Before running Algorithm 1, the distance field on M isconstructed in O(n 2 log n) time [45] <strong>and</strong> an interval tree isconstructed for all the triangles in M in O(n ′ log n ′ ) time.Since multiple point sources behave like pseudo-sources,the construction <strong>of</strong> iso-<strong>contours</strong> <strong>of</strong> multiple sources isidentical to that <strong>of</strong> a single source.5.2 <strong>Bisectors</strong>According to Definition 3, the singular loci <strong>of</strong> a multisourcegeodesic distance field contain the trimmed bisectorsthat contribute to the Vornoi diagram <strong>of</strong> the pointsource set P on M. In this section, we explicitly constructthe bisector <strong>of</strong> two source points p <strong>and</strong> q on M.Property 9: The bisector B(p, q) goes through an edgee at the location that delimits two visibility wedges at e<strong>and</strong> the two wedges have the source point ID Id pt = p<strong>and</strong> Id pt = q, respectively.Preprocess. In the worst case, one bisector can gothrough a triangle as many as n times [48], [40].Similar to iso-contour construction, the mesh modelis also preprocessed to avoid this hypothetical worstcasecomplexity. We first examine each edge in M<strong>and</strong> guarantee that each edge contains at most oneintersection point with the bisector: if this is not thecase, the edge is subdivided. Denote the number <strong>of</strong>triangles in a preprocessed mesh by n ′ . Starting fromp <strong>and</strong> q, we propagate the visibility wedges in acontinuous-Dijkstra fashion. During propagation, weuse a list to store those edges at which a visibilitywedge is updated with a neighbor wedge belongingto a different source. To the end, the edge list isconverted into a queue Que which contains all thetriangles through which the bisector passes. Similar to
IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, VOL. 33, NO. 8, PAGE 1502-1517, AUGUST 2011 9Fig. 16. The <strong>Voronoi</strong> diagrams <strong>of</strong> 30 r<strong>and</strong>omly generated points on different models <strong>of</strong> genus-0. The statistical data issummarized in Table 1.iso-<strong>contours</strong>, the following marching algorithm is usedto construct the bisector on a general genus-r model.Algorithm 2: genus_r_bisector(M, p, q)Input. A preprocessed mesh M <strong>and</strong> two point sites p, q.Output. The requested bisector with the number <strong>of</strong>closed curves.1. Build the distance field on M with point sites p, q<strong>and</strong> output the queue Que;2. Set curve number bn = 0;3. While (Que is not empty)3.1. bn = bn + 1;3.2. Pop the first element t <strong>of</strong> Que;3.3. Marching triangles with the initial triangle t;3.4. Remove all the marched triangles from Que.Property 10: Algorithm 2 runs in O(n 2 log n) time.The pro<strong>of</strong> <strong>of</strong> Property 10 is sketched as follows. Ittakes O(n 2 log n) time to compute the geodesic distancefield [45] <strong>and</strong> report all triangles containing the bisectorB(p, q) with the triangle number k < n ′ < n 2 . Themarching process takes O(k) time <strong>and</strong> the queue Queoperations in O(k log k) time. So the total running timeis O(n 2 log n).5.3 <strong>Voronoi</strong> <strong>Diagrams</strong>The preprocess step for constructing <strong>Voronoi</strong> diagramson M is the same as that for the bisector construction.Also similar to the bisector case, we record a list to storethose edges at which a visibility wedge is updated witha neighbor wedge belonging to a different source. Afterthe geodesic distance field construction, the list <strong>of</strong> edgesLE is converted into a list <strong>of</strong> triangles LT incident to LE.From Definition 6 <strong>and</strong> Property 9, the following propertyholds.Property 11: If any triangle in LT has all its three edgescontained in LE, then it contains a branch point. Eachtriangle in LT that does not contain a branch point ispassed through by a single piece <strong>of</strong> a bisector.Property 11 gives us a valuable means to computethe analytic structure <strong>of</strong> the <strong>Voronoi</strong> diagram on M.We separate the list LT into two sub-lists. One is LBTwhose elements contain branching points. The other isLT T = LT \LBT . The following algorithm constructsthe <strong>Voronoi</strong> diagram <strong>of</strong> point set P on M using thedata structure as shown in Figures 9 <strong>and</strong> 11.Algorithm 3: genus_r_<strong>Voronoi</strong>_diagram(M, P )Input. A preprocessed mesh M <strong>and</strong> a point site set P .Output. The requested <strong>Voronoi</strong> diagram using the datastructure depicted in Figures 9 <strong>and</strong> 11.1. Build the distance field on M with the set P <strong>and</strong>output the lists LE <strong>and</strong> LT ;2. Separate LT T into LBT <strong>and</strong> LT T ;3. Create a branch-point list BP : each point bp i withsources (s i (1), s i (2), s i (3)) corresponds to a trianglet i in LBT .4. For all the branch points bp i ∈ BP4.1. For m = 1 to 34.1.1. If the bisector B(s i (m), s i ((m + 1)%3)) is notcomputed4.1.1.1. Marching B(s i (m), s i ((m + 1)%3)) startedfrom t i <strong>and</strong> ended at another t j in LBT .4.1.1.2. Remove all marched triangles from LT T .5. While (LT T is not empty) //some bisectors do not//have branch points by Definition 65.1. Create a new entry in the bisector list;5.2. Pop one element t in LT T ;5.3. Marching triangles with the initial triangle t;5.4. Remove all the marched triangles from LT T .Property 12: Algorithm 3 runs in O(n 2 log n) time.The pro<strong>of</strong> <strong>of</strong> Property 12 is sketched as follows. It takesO(n 2 log n) time to compute the geodesic distance field[45] <strong>and</strong> report the set LE, LBT <strong>and</strong> LT T . Let k be thenumber <strong>of</strong> triangles passed by the boundaries <strong>of</strong> V D(P ).The two loops in Steps 4 <strong>and</strong> 5 take time O(k log k). Sothe total running time is O(n 2 log n).
Page 4 and 5: IEEE TRANSACTIONS ON PATTERN ANALYS
Page 14: IEEE TRANSACTIONS ON PATTERN ANALYS

Construction of Iso-contours, Bisectors and Voronoi Diagrams on ...

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?